Knot Floer homology of satellite knots with (1,1) patterns

For pattern knots admitting genus-one doubly pointed bordered Heegaard diagrams, we give an immersed-curve approach to compute the knot Floer chain complexes of the corresponding satellite knots. In particular, this approach provides a convenient way to compute the knot concordance invariant t. For patterns P obtained from two-bridge links b(p, q), we derive a formula for the t-invariant of P(T-2,T-3) and P(-T-2,T-3) in terms of (p, q). We use this formula to study whether such patterns induce homomorphisms on the knot concordance group, providing a glimpse at a conjecture due to Hedden.